He confessed that it was a common girl and that Tywin made him lie. A commoner wasn't a fit match for a Lannister of The Rock, so Tywin devised a plan to make sure Tyrion never considered marrying spontaneously again. He threatened the girl and made her out to be a whore. Jaime felt ashamed because he let their father ruin Tyrion's one chance at true love. When he tells Tyrion in the dungeon just before setting him free, Tyrion lies and says that he did indeed kill Joff. This goes on to haunt Jaime, who thinks that he traded his fathers life for the life of the man who killed his son. When Jaime reveals the long kept secret Tyrion is finally unhinged, and after all the years of suppressed hatred of his family he finally snaps. He kills Tywin and later swears that he'll kill all his kin. Even Jaime, to whom he actually had quite a good relationship up to this point Major spoiler if you haven't read all the books!

Again spoilers about why Tyrion was in the place mentioned earlier: Yes, it's going to be in season 4. Tyrion is convicted of killing Joff during his wedding. Said wedding and scenes from his trial (trial by battle) can be seen in the season 4 teaser. Shit's about to get real!